IN THE FOLD: The president won the allegiance of the final Dem health-bill holdout, Sen. Ben Nelson , seen in snowy DC yesterday.AP

WASHINGTON — After weeks of bitter fighting and backdoor deals, Senate Democrats announced yesterday that they’d finally secured enough votes to pass a sweeping, $871 billion overhaul of the health-care system that aims to insure 30 million Americans.

The lone holdout, Sen. Ben Nelson (D-Neb.), said he was satisfied at last that the bill ensured no federal funds would go to fund abortion.

In addition to including stricter abortion-funding language, Senate Majority Leader Harry Reid sweetened the deal for Nelson by guaranteeing that his state would receive full federal funding for Medicaid costs, an estimated $45 million surplus over the next decade.

The new rules give states the choice of opting out of the newly created federal insurance exchanges if they provide abortions.

Prominent pro-life advocates such as House Republican leader John Boehner and the National Right to Life committee denounced the legislation for not doing enough to stop federal money from going to pay for elective abortions.

Pro-choice advocates such as like Planned Parenthood and the National Organization for Women criticized the bill for unfairly excluding women’s reproductive rights.

GOP leaders also objected to a slew of new taxes, including a 10 percent tax on tanning-salon visits and a 0.9 percentage-point increase on the Medicare payroll tax for people making more than $200,000 a year.